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(Solved): The cut condition suggests a family of algorithms determine a minimum cost-spanning tree from a weig ...



The cut condition suggests a family of algorithms determine a minimum cost-spanning tree from a weighted and connected graph (G, c). This family of algorithms maintains an acyclic subgraph F and at each iteration, a minimum cost edge of a disjoint cut of F is added to F, until F is a spanning tree. Consider the following idea. The algorithm maintains at each iteration a connected subgraph F of G. The first iteration starts with F = E. Each iteration consists of the following. If F is a spanning tree, then return F and stop. If F is not a spanning tree, then select an edge e of F that is safe. Start a new iteration with F ? {e} in place of F. Show how to define the notion of safe such that:
(i) it is possible to determine in linear time in \left | V_{F} \right | + \left | E _{F} \right | if an edge is safe, and
(ii) the returned spanning tree is of minimal cost.



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i) The algorithm terminates with a spanning
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